The concept of uniform integrability is an important concept in functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

 and probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

.

If is a finite measure, a subset is said to be uniformly integrable if


Rephrased with a probabilistic language, the definition becomes : a family of integrable random variables is uniformly integrable if
This definition is useful in limit theorems, such as the Vitali convergence theorem

Vitali convergence theorem

In mathematics, the Vitali convergence theorem is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is useful when a dominating function cannot be found for the sequence of functions in question; when such a dominating function can be found, Lebesgue's theorem...

.

Sufficient conditions

• If , the singleton is uniformly integrable, as an easy consequence of Lebesgue's dominated convergence theorem
  Dominated convergence theorem
  In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
  
  . Similarly, finite subsets of are uniformly integrable.

• If , the set is uniformly integrable.

• A set bounded in is uniformly integrable.

Bounded and absolutely continuous

A subset is uniformly integrable iff

IFF

IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...

 it is uniformly bounded (i.e. ) and absolutely continuous, i.e. for any there exists so that
.

Dunford

Nelson Dunford

Nelson Dunford was an American mathematician, known for his work in functional analysis, namely integration of vector valued functions, ergodic theory, and linear operators. The Dunford decomposition, Dunford–Pettis property, and Dunford-Schwartz theorem bear his name.He studied mathematics at the...

–Pettis

Billy James Pettis

Billy James Pettis , was an American mathematician, known for his contributions to functional analysis.-See also:*Dunford–Pettis property*Dunford–Pettis theorem*Milman–Pettis theorem*Orlicz–Pettis theorem*Pettis integral...

 theorem

A subset is uniformly integrable if and only if it is relatively compact for the weak topology

Weak topology

In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...

.

de la Vallée-Poussin

Charles Jean de la Vallée-Poussin

Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin was a Belgian mathematician. He is most well known for proving the Prime number theorem.The king of Belgium ennobled him with the title of baron.-Biography:...

 theorem

The family is uniformly integrable iff

IFF

IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...

 there exists a nonnegative increasing convex function such that and

Relations to convergence of random variables

Convergence of random variables

In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes...

• A sequence converges to in the norm if and only if it converges in measure
  Convergence in measure
  Convergence in measure can refer to two distinct mathematical concepts which both generalizethe concept of convergence in probability.-Definitions:...
  
   to and it is uniformly integrable.

• In probabilistic language, a sequence of integrable random variables converges to in mean if and only if it converges in probability to and it is uniformly integrable.